Chapter 8 - Test - Page 412: 15

$x = 2~cos~2t$ $y = 2~sin~2t$ $t$ in $[0,2\pi]$ We can see the graph below:

Work Step by Step

$x = 2~cos~2t$ $y = 2~sin~2t$ $t$ in $[0,2\pi]$ When $t = 0$: $x = 2~cos~0 = 2$ $y = 2~sin~0 = 0$ When $t = \frac{\pi}{8}$: $x = 2~cos~\frac{\pi}{4} = 2~\frac{\sqrt{2}}{2} = \sqrt{2}$ $y = 2~sin~\frac{\pi}{4} = 2~\frac{\sqrt{2}}{2} = \sqrt{2}$ When $t = \frac{\pi}{4}$: $x = 2~cos~\frac{\pi}{2} = 0$ $y = 2~sin~\frac{\pi}{2} = 2$ When $t = \frac{\pi}{3}$: $x = 2~cos~\frac{2\pi}{3} = -1$ $y = 2~sin~\frac{2\pi}{3} = \sqrt{3}$ When $t = \frac{\pi}{2}$: $x = 2~cos~\pi = -2$ $y = 2~sin~\pi = 0$ When $t = \frac{3\pi}{4}$: $x = 2~cos~\frac{3\pi}{2} = 0$ $y = 2~sin~\frac{3\pi}{2} = -2$ When $t = \pi$: $x = 2~cos~2\pi = 2$ $y = 2~sin~2\pi = 0$ As $t$ increases from $t = \pi$ to $t=2\pi$, the points around the circle of radius 2 are repeated. We can see the graph below:

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